Chapter 8 Robust Designs

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Chapter 8Robust Designs
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Robust Designs
Focus A Few Primary Factors Output
Best/Robust Settings
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Robust Designs
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Robust Experimentation

• Try to reduce the variation in the
  product.process, not just set it on target.
• Try to find conditions of design factors which
  make the product/process insensitive or robust to
  its environment.
• The idea was championed by Genichi Taguchi.
• Taguchi really opened a whole area that
  previously had been talked about only by a few
  very applied people.
• His methodology is heavily dependent on design of
  experiments, but he wanted to look at not just
  the mean but also the variance.

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Classification of Factors

• Control FactorsDesign factors that are to be set
  at optimal levels to improve quality and reduce
  sensitivity to noise
• Dimensions of parts, type of material, etc
• Noise FactorsFactors that represent the noise
  that is expected in production or in use
• Dimensional variation
• Operating Temperature
• Adjustment Factor Affects the mean but not the
  variance of a response
• Deposition time in silicon wafer fabrication
• Signal Factors Set by the user to communicate
  desires of the user
• Position of the gas pedal

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Example Ignition Coil

• Most automotive sub-assemblies like the
  alternator, the ignition coil, and the electronic
  control module must undergo testing to determine
  if they are resistant to salt water that may be
  splashed on them from the read. An automotive
  supplier is testing an ignition coil to determine
  if it will withstand salt water. The following
  factors are tested

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Control or Noise Factor?
Factor Low Level High Level Factor Type
Housing Material Polyethylene (9/lb) Polypropylen (20/lb) C
Concentration Of Salt 10 20 N
Water Temperature 5ºC 15ºC N
Water Pressure 10 PSI 20 PSI N
Seal Material Silicone Vinyl C
Seal Thickness 0.02 0.03 C
Exposure Time 1 hr. 5 hrs. N
Type of Salt Detroit Blue Chicago Pink N
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Taguchis Design of Experiments Ideas

• Use Orthogonal Arrays (e.g., Resolution III
  Fractional Factorials) that only estimate main
  effects and specified interactions.
• Intentionally vary the noise factors so that you
  choose a set of conditions that will work well in
  the face of the noise expected in the actual
  application.
• Inner/Outer Arrays
• Two statistical designs per experiment
• A design in the control factors called an inner
  array
• A design in the noise factors called an outer
  array
• Good for estimating CxN Factor Interactions.

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Taguchis Analysis IdeasSignal to Noise Ratio

• A single response which makes a tradeoff between
  setting the mean to a desirable level while
  keeping the variance low.
• Always try to MAXIMIZE a SN Ratio
• There are three types
• Smaller is Better
• Target is Best
• Larger is Better

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Smaller is Better

• Maximize the signal to noise ratio
• Run a confirmatory experiment

The signal to noise ratio confounds the mean and
the variance together and assumes that the
variance is proportional to the mean.
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Examples with the same SNS Ratio
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Target is Best

• Maximize the signal to noise ratio
• Adjust the mean to target using an adjustment
  factor that has no effect on the signal to noise
  ratio

The signal to noise ratio does tend to prefer
combinations of levels that decrease noise and
are on target, but assumes that the variance is
proportional to the mean. If it isnt, then the
method gives an unknown compromise.
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Larger is Better

• Maximize the signal to noise ratio
• Run a confirmation

The signal to noise ratio again confounds the
mean and the variance together and assumes that
the variance is proportional to the mean.
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Sources of Variance

• Measurement
• Setup
• Blocks
• Noise Factors
• Manufacturing Variation
• Field Conditions

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Repeats
?2 represents measurement error
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Replicates
?2 represents measurement and setup error
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Blocks
?2 represents variance between blocks and
measurement and setup error.
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Noise Factors
?2 represents variance expected in the field due
to field conditions and measurement and setup
error. The idea of robustness is to set the xs
to minimize the variance, while keeping the
average response at a desired level.
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I/O (crossed) ArrayAn Inner Array of Control
Factors and an Outer Array of Noise Factors
?2 represents variance expected in the field and
measurement and setup error.
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Analysis

• Taguchi suggests SN Ratios.
• We could analyze the average and the standard
  deviation separately.
• We could define some other summary response
• Minimize the range or the curvature
• We could include effects for each y and explain
  each one individually like we do with blocks.

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Analyzing the Mean and Standard Deviation
Separately

• For each row of ys, we can take the mean of the
  row as one response and the log of the standard
  deviation as another response.
• The log is used because the variance of the
  standard deviation is not constant as the
  standard deviation increases.
• We then try to compromise between setting the
  mean to a desirable level and reducing the
  standard deviation.
• Effects on the mean are called Location Effects.
  
• Effects on the standard deviation are called
  Dispersion Effects

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Analysis of I/O (Crossed) Arrays

• When we have I/O array there will be three types
  of effects
• Control Main Effects and Interactions
• Noise Main Effects and Interactions
• Control by Noise Interactions
• Depending on HOW WE RUN THE EXPERIMENT, the
  different effects may have different variances.
• Suppose that there are n runs in the control
  array and m runs in the noise array. There will
  be a total of nm data points.

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Example Cake Mix Example

• Amounts of ingredients like egg, shortening and
  flour are the control factors (n levels in the
  control array)
• Time and Temperature in baking are the noise
  factors (m levels in the noise array)

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How do we run the experiment?

1. Fully Randomized
2. Split Plot
3. Strip Block

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1. Fully Randomized

• Make nm products and test them each separately.

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2. Split Plot

• Experiments where you cant or dont think it is
  worth it to fully randomize the experiment
• Used when there are certain effects that you want
  to estimate with high precision. (like blocking)
• Often needed when some factors are hard to
  change and others are easy to change
• I/O arrays are often but not always run as split
  plot experiments

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Split Plot Variance

• Whole-plot effects have high variance.
• Sub-plot effects have lower variance.
• Only effects that have the same variance should
  be put on the same normal plot.

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2a. Split Plot ArrangementControl Factors as
Whole-Plot Effects

• Mix up n big batches of batter
• Split each batch into m parts
• Bake m x n cakes in m x n oven runs
• All Control Effects have error due to batch error
  and due to random error?
• All Noise Effects and CxN interactions have only
  random error?
• 2 Normal Plots

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2b. Split Plot ArrangementControl Factors as
Sub-Plot Effects

• Mix up m x n batches of batter
• Bake all n batter types together under each set
  of noise conditions
• Bake m x n cakes in m oven runs
• All Noise Effects have error due to oven error
  and due to random error?
• All Control Effects and CxN interactions have
  only random error?
• 2 Normal Plots

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3. Strip Block Arrangement

• Mix up n batches of batter
• Split each batch into m parts
• Bake all n batter types together under each set
  of noise conditions
• Bake m x n cakes in m oven runs
• All Noise Effects have error due to oven error
  and due to random error?
• All Control Effects have error due to batch error
  and due to random error?
• CxN interactions have only random error?
• 3 Normal Plots

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The purpose of each of these configurations is to
reduce the experimental cost and target certain
effects to have higher precision of measurement.
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HOW WE RUN THE EXPERIMENT

• Fully Randomize
• Make nm different products and test them each
  separately
• One normal plot with all effects
• Split Plot
• Make n different products and test each product
  in nm separate tests.
• Two normal plots (1) Control (2) Noise and CxN
• Make nm different products and test them in
  groups of n in m tests.
• Two normal plots (1) Control and CxN (2) Noise
• Strip Block
• Make n different products and test them each in
  m tests.
• Three normal plots (1) Control (2) CxN (3) Noise

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Control by Noise Factor Interactions
Interaction between Shortening and Temperature
Shortening
Moistness
Set Control Factor (e.g. Shortening) to Low Level
Noise Factor (e.g. Temp.)
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Control by Noise Factor Interactions
Interaction between Flour and Temperature
Flour
Moistness
Set Control Factor (e.g. Flour) to Middle Level
Noise Factor (e.g. Temp.)
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Control by Noise Factor Interactions
Interaction between Egg and Temperature
Egg
Moistness
Set the Adjustment Factor (e.g. Egg) to Set
Mean on Target
Noise Factor (e.g. Temp.)
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I/O Arrays and Control by Noise Interactions

• I/O arrays always estimate the CxN Interactions
  with the highest precision (lowest variance).
• Sometimes the I/O array design is larger than it
  needs to be to get the same resolution
• However, if we are interested in estimating CxN
  interactions, then an I/O array design can be a
  very good design.

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Confounding in I/O Arrays

• Same as in standard arrays
• Write down identity relationships from generators
• Multiply out all possible combinations to get
  defining relation
• Multiply defining relation by any effect to get
  alias structure
• Equivalently we could get the individual defining
  relations and multiply every possible pair of
  terms
• Resolution of a crossed array experiment is the
  minimum of the resolutions of the individual
  arrays
• RC Resolution of Control Array
• RN Resolution of Noise Array

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Defining Relation IABCPQRABCPQR
Notice that as long as each individual array is
Resolution III the shortest word containing both
control and noise factors will have length 6.
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Confounding Pattern

• BPACPBQRACQR
• BQACQBPRACPR
• BRACRBPQACPQ
• CPABPCQRBCQR
• CQABQCPRABPR
• CRABRCPQABPQ

• ABCAPQRBCPQR
• BACBPQRACPQR
• CABCPQRABPQR
• PABCPQRABCQR
• QABCQPRABCPR
• RABCRPQABCPQ
• APBCPAQRBCQR
• AQBCQAPRBCPR
• ARBCRAPQBCPQ

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Control by Noise Factor Interactions will be
confounded with interactions of order Min(RC,
RN). So, if both arrays have Resolution III
then CxN interactions will be confounded with
three factor interactions.
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Summary

• I/O arrays are always good for studying CxN
  interactions
• I/O arrays are often but not always run as split
  plot experiments
• Split plot is determined by how you run the
  experiment not how you design it.
• You must know how the experiment was run in order
  to analyze it with the correct error structure

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My Recommendations

• Think about the idea of robustness in the
  planning stage of your experiment and choose the
  response accordingly.
• If you choose to reduce variance, then make sure
  that the variance that you are reducing is the
  variance that you want to reduce and that it is
  worth the experimental effort.
• Analyze the mean and variance separately and make
  decisions concerning the tradeoffs between
  reducing variance and achieving the desired mean
  value.
• Use Inner and Outer Arrays to estimate Control by
  Noise Interactions for designing Robust Products
  and Processes.

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Assembled Designs
A class of experimental design for estimating
location effects, dispersion effects, and
variance components.
Parameters r of Design Points n of
observations per Treatment Combination (assume
constant) s of different structures used in
the design BT total number of Batches
r 8 n 4 s 2 BT 20
x2
x3
x1
Works for anything that involves making batches
and taking observations
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Notation for Assembled Designs
Example 23 design, Splitting Generator ABC,
r8, s2, B14, B23, n6